The inverse tangent function f(x)=arctanx is approximated near x=1 by the third–degree Taylor polynomial P3 centered at x=1. The approximation will be used for inputs in the interval [1,1.20].
(a) Find P3(x) for f centered at x=1. Show all derivatives used.
(b) Show that for every z in [1,1.20], the fourth derivative satisfies ∣f(4)(z)∣≤0.40. Provide complete reasoning.
(c) Employ the bound from part (b) to obtain a numerical upper bound for ∣f(1.20)−P3(1.20)∣. Give your answer to five decimal places.